recurrence tutorial: laminar vortex shedding

This 2D case of vortex shedding in laminar cross-flow demonstrates the
concept of recurrence.

This is a bit work-in-progress: please check whether this tutorial runs with
the recurrence model and tools of CFDEMcoupling, namely rStatAnalysis.

The simulation roughly goes through three stages:
  * The initial solution computed by potentialFoam
  * A period of symmetric, steady-state flow
  * Finally, periodic vortex shedding

These three stages are clearly visible in the recurrence plot.

  * We see how not one of the later velocity fields is similar to the initial one
  * We see the intermediate stage with a symmetric flow field
  * We see the periodic vortex shedding

tutorials/rStatAnalysis/laminarFlowOvercylinder/0.org/U

@@ -0,0 +1,51 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  4.0                                   |
+|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+    version     2.0;
+    format      binary;
+    class       volVectorField;
+    location    "0";
+    object      U;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+dimensions      [0 1 -1 0 0 0 0];
+
+internalField   uniform (0 0 0);
+
+boundaryField
+{
+    cylinder
+    {
+        type            fixedValue;
+        value           uniform (0 0 0);
+    }
+    inlet
+    {
+        type            fixedValue;
+        value           uniform (0.0040120 0 0); // Re = 200
+    }
+    outlet
+    {
+        type            inletOutlet;
+        inletValue      uniform (0 0 0);
+        value           uniform (0 0 0);
+    }
+    "side.*"
+    {
+        type            slip;
+    }
+    frontBack
+    {
+        type            empty;
+    }
+}
+
+
+// ************************************************************************* //

tutorials/rStatAnalysis/laminarFlowOvercylinder/0.org/p

@@ -0,0 +1,48 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  4.0                                   |
+|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+    version     2.0;
+    format      binary;
+    class       volScalarField;
+    location    "0";
+    object      p;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+dimensions      [0 2 -2 0 0 0 0];
+
+internalField   uniform 0;
+
+boundaryField
+{
+    cylinder
+    {
+        type            zeroGradient;
+    }
+    inlet
+    {
+        type            zeroGradient;
+    }
+    outlet
+    {
+        type            fixedValue;
+        value           uniform 0;
+    }
+    "side.*"
+    {
+        type            zeroGradient;
+    }
+    frontBack
+    {
+        type            empty;
+    }
+}
+
+
+// ************************************************************************* //

tutorials/rStatAnalysis/laminarFlowOvercylinder/Allclean

@@ -0,0 +1,23 @@
+#!/bin/bash
+#
+# clean the case
+
+cd ${0%/*} || exit 1    # Run from this directory
+
+# Source tutorial clean functions
+. $WM_PROJECT_DIR/bin/tools/CleanFunctions
+
+# remove mesh
+rm -rf constant/polyMesh/*
+
+rm -rf 0 > /dev/null 2>&1
+
+cleanCase
+
+rm rec_jump.dat
+rm recurrenceMatrix
+rm recurrencePath
+
+cd recMat
+
+./clean.sh

tutorials/rStatAnalysis/laminarFlowOvercylinder/Allrun

@@ -0,0 +1,9 @@
+#!/bin/bash
+#
+# run the case
+
+./meshIt
+
+./run
+
+./post

tutorials/rStatAnalysis/laminarFlowOvercylinder/ReadMe.md

@@ -0,0 +1,157 @@
+# Laminar flow over cylinder - recurrent flow
+
+This OpenFOAM case demonstrates the recurring flow patterns of vortex sheding
+of a cylinder in laminar cross flow. Apart from judging the recurrence using
+the eyeball norm, i.e. by looking at the solution, we are computing the
+recurrence matrix from the computed time steps.
+
+The recurrence plot [1], a graphical representation of the recurrence matrix,
+allows us to clearly distinguish between the initial phase of the solution and
+the quasi steady state, which exhibits a nice periodic flow.
+
+
+## Case description
+
+The mesh is created with blockMesh using a simple blocking strategy with 6
+blocks. The domain is meshed using one cell in z direction, thus the flow
+simulation is a 2D simulation. The patch on the left acts as inlet, the patch
+on the right is an outlet. The top and bottom patches are frictionless walls.
+The cylindrical void in the domain employs wall boundary conditions.
+
+
+## Results
+
+The image below, is the recurrence plot of this simulation. The recurrence plot
+is based on the recurrence matrix, which in turn was computed from the velocity
+fields of this simulation.
+
+Each entry in the recurrence matrix is computed by comparing two states of the
+simulation, i.e. time steps. In this case the velocity fields of all written
+time steps were compared to each other. For each entry, the magnitude of the
+difference of two velocity fields was computed. This magnitude, which itself is
+still a field quantity was summed up over all cell values, and finally
+normalized. Consequently, the values along the main diagonal, i.e. both indices
+for x and y, are zero, as the corresponding state is compared to itself.
+
+![The recurrence plot](doc/distMatrix.png)
+
+As the simulation is initialized using *potentialFoam*, we have an initial
+velocity field, which is closer to reality than a trivial initial field, i.e.
+zero velocity everywhere. However, the initial velocity field is quite
+unphysical as it contains artifacts due to the mesh, as shown below.
+Notice, in the recurrence plot, the red bands along the x-axis for low values
+of y, and along the y-axis for low values of x. This is a result of the initial
+velocity field bearing no resemblance to all subsequent velocity fields.
+Thus, a horizonal or vertical high-value band is the result of a state, which
+does not re-occur.
+
+Thus, a low value indicates great similarity between two time steps. E.g. the
+value at (500,1000) is very low (shown in blue in the plot), which means that
+the velocity fields at the time 500 s and 1000 s are very similar, as shown
+below.
+The whole blue square in the recurrence plot indicates a certain period in
+which the solution changes very little over time. Thus, one could assume, if
+the simulation were to stop aroung t = 1000 s, that the flow field in the
+simulation had entered its steady state.
+
+However, as the simulation progresses, vortex shedding kicks in, and we end up
+having a nice, periodic solution. This is indicated by the narrow bands
+parallel to the main diagonal in the upper right corner of the recurrence plot.
+A low-value band parallel to the main diagonal means that the same sequence of
+states can be observed with a certain time shift. A purely periodic system will
+show the exactly same state after completing a full cycle.
+
+The narrow bands in this recurrence plot strongly suggests that the laminar
+vortex shedding exhibits periodic behaviour.
+
+
+### Initial velocity field
+
+![A snapshot of the flow](doc/vel000.png)
+t = 0 s
+
+### Velocity fields at t = 500 s and t = 1000 s
+
+The following two velocity fields are very similar to each other. In the
+recurrence matrix the value for indicating the similarity of these two velocity
+fields is 0.001.
+
+![A snapshot of the flow](doc/vel050.png)
+t = 500 s
+
+![A snapshot of the flow](doc/vel100.png)
+t = 1000 s
+
+### Velocity fields at t = 1500 s, t = 1850 s and t = 1950 s
+
+The following two velocity fields are very similar to each other. In the
+recurrence matrix the value for indicating the similarity of these two velocity
+fields is 0.047.
+
+![A snapshot of the flow](doc/vel150.png)
+t = 1500 s
+
+![A snapshot of the flow](doc/vel185.png)
+t = 1850 s
+
+The following velocity field is very dissimilar to the preceeding two, although
+it looks very alike. However, whereas the fields at t = 1500 s and t = 1850 s
+are nearly at the same part of the cycle, these two fields and the field at
+t = 1950 s are on opposite sides of the cycle. The value of the recurrence
+matrix, which indicates similarity is 0.496, respectively 0.488.
+
+![A snapshot of the flow](doc/vel195.png)
+t = 1950 s
+
+
+### Additional notes
+
+Apart from running the meshing tools *blockMesh* and *renumberMesh*, the solvers
+*pimpleFoam* and *potentialFoam*, this case runs the tool *rStatAnalysis*, which
+is part of the recurrence model.
+
+
+## Post-processing
+
+This tutorial uses a number of third party software tools for post-processing.
+
+[Octave](https://www.gnu.org/software/octave/) is open source under the GPL and
+is available for all platforms. Octave is widely compatible to MATLAB, so you
+can run the post-processing script also with MATLAB.
+
+[gnuplot](http://www.gnuplot.info/) is a portable command-line driven graphing
+utility for Linux, MS Windows, OSX, and many other platforms. It is open source
+under its own licence.
+
+[TeX Live](https://www.tug.org/texlive/) is an easy way to get up and running
+with the TeX document production system. It provides a comprehensive TeX system
+with binaries for most flavors of Unix, including GNU/Linux, and also Windows.
+It includes all the major TeX-related programs, macro packages, and fonts that
+are free software, including support for many languages around the world.
+
+[pdfCrop](https://ctan.org/pkg/pdfcrop) is part of a larger collection
+[texlive-extra-utils](https://launchpad.net/ubuntu/+source/texlive-extra) of
+auxiliary tools for the TeX system. This tool, however, is non essential to
+the functioning of this tutorial. If it is not present or available on your
+system, simply comment out the call to it.
+
+The post-processing script has been tested with:
+
+* GNU Octave, version 4.0.0
+* gnuplot 5.0
+* pdfTeX 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian)
+* texlive-extra-utils 2015.20160320-1
+
+
+## Tested
+
+This collection of cases has been tested with:
+
+* OpenFOAM-5.0
+
+
+## References
+
+[1] T. Lichtenegger and S. Pirker. Recurrence CFD – A novel approach to simulate
+multiphase flows with strongly separated time scales. Chemical Engineering Science,
+153:394-410, 2016.

tutorials/rStatAnalysis/laminarFlowOvercylinder/constant/recProperties

@@ -0,0 +1,82 @@
+/*---------------------------------------------------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  1.4                                   |
+|   \\  /    A nd           | Web:      http://www.openfoam.org               |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+
+
+FoamFile
+{
+    version         2.0;
+    format          ascii;
+
+    root            "";
+    case            "";
+    instance        "";
+    local           "";
+
+    class           dictionary;
+    object          recProperties;
+}
+
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+//===========================================================================//
+// sub-models & settings
+
+
+recModel standardRecModel;
+
+verbose false;
+
+recNorm sqrDiffNorm;
+
+recPath simpleRandomPath;
+
+
+volScalarFields
+(
+);
+
+volVectorFields
+(
+    U
+);
+
+surfaceScalarFields
+(
+);
+
+//verbose;
+
+
+
+//===========================================================================//
+// sub-model properties
+
+standardRecModelProps
+{
+    dataBase    ".";
+    
+    skipZero    on;
+}
+
+sqrDiffNormProps
+{
+  fieldType "volVectorField";
+  fieldName U;
+  
+  //verbose   false;
+}
+
+simpleRandomPathProps
+{
+}
+
+noRecStatAnalysis
+{
+}
+
+// ************************************************************************* //

tutorials/rStatAnalysis/laminarFlowOvercylinder/constant/transportProperties

@@ -0,0 +1,25 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  2.3.0                                 |
+|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+    version     2.0;
+    format      ascii;
+    class       dictionary;
+    location    "constant";
+    object      transportProperties;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+transportModel  Newtonian;
+
+// water, 20°C, VDI Wärmeatlas
+nu              nu [ 0 2 -1 0 0 0 0 ] 1.003e-06;
+
+
+
+// ************************************************************************* //

tutorials/rStatAnalysis/laminarFlowOvercylinder/constant/turbulenceProperties

@@ -0,0 +1,20 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  2.3.0                                 |
+|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+    version     2.0;
+    format      ascii;
+    class       dictionary;
+    location    "constant";
+    object      turbulenceProperties;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+simulationType laminar;
+
+// ************************************************************************* //

tutorials/rStatAnalysis/laminarFlowOvercylinder/meshIt

@@ -0,0 +1,6 @@
+#!/bin/bash
+
+blockMesh 2>&1 | tee log.blockMesh 2>&1
+
+# renumber the mesh
+renumberMesh -overwrite 2>&1 | tee log.renumberMesh 2>&1

tutorials/rStatAnalysis/laminarFlowOvercylinder/post

@@ -0,0 +1,10 @@
+#!/bin/bash
+#
+# run the post-processing
+
+rStatAnalysis 2>&1 | tee log.rStatAnalysis 2>&1
+
+cd recMat
+
+./plot.sh
+

tutorials/rStatAnalysis/laminarFlowOvercylinder/recMat/clean.sh

@@ -0,0 +1,7 @@
+#!/bin/bash
+
+rm -f *.log
+rm -f *.aux
+rm -f *.txt
+rm -f *.pdf
+rm -f myDistMatrix.tex

tutorials/rStatAnalysis/laminarFlowOvercylinder/recMat/getRecMat.m

@@ -0,0 +1,75 @@
+% open files
+fid1 = fopen('../recurrenceMatrix');
+
+% read dimensions
+A1 = fscanf(fid1, '%g %g');
+
+% skip the first two lines
+tline = fgetl(fid1);
+tline = fgetl(fid1);
+
+% get dimensions
+N1 = A1(1)
+M1 = A1(2)
+
+% allocate space
+B0 = zeros(M1,N1);
+
+% read data
+for i=1:N1
+    B0(:,i) = fscanf(fid1, '%g', inf);
+    tline = fgetl(fid1);
+    tline = fgetl(fid1);
+end
+
+% close files
+fclose(fid1);
+
+
+% skip this many leading entries
+sle = 0;
+
+B1 = zeros(M1-sle,N1-sle);
+
+B1 = B0(1+sle:M1,1+sle:N1);
+
+maxval=0.0;
+%for i=1:M1
+%  for j=1:N1
+%    if(B1(i,j)>maxval)
+%      maxval=B1(i,j);
+%    endif
+%  end
+%end
+
+maxval = 1.0;
+for i=1:M1-sle
+  for j=1:N1-sle
+    B1(i,j)=1-B1(i,j)/maxval;
+   % B1(i,j)=B1(i,j)/maxval;
+  end
+end
+
+
+
+% write full matrix to simple text file
+dlmwrite('myMatrix.txt',B1,'delimiter','\t','precision',3)
+
+
+
+%plot(C1)
+%saveas(gcf,'Plot','png')
+
+%B1=B1*1;
+
+
+%hold on
+
+
+%colormap(jet(50))
+%imagesc(B1)
+%colorbar
+
+%saveas(gcf,'Figure','png')
+
+

tutorials/rStatAnalysis/laminarFlowOvercylinder/recMat/makePlot.tex

@@ -0,0 +1,7 @@
+\documentclass[preview]{standalone}
+\usepackage{graphicx}
+\begin{document}
+\begin{figure}
+ \input{myDistMatrix}
+\end{figure}
+\end{document}

tutorials/rStatAnalysis/laminarFlowOvercylinder/recMat/plot.sh

@@ -0,0 +1,4 @@
+octave getRecMat.m
+gnuplot plotDistMat.gnu
+pdflatex makePlot.tex
+pdfcrop makePlot.pdf distMatrix.pdf

tutorials/rStatAnalysis/laminarFlowOvercylinder/recMat/plotDistMat.gnu

@@ -0,0 +1,48 @@
+reset
+clear
+
+set terminal cairolatex pdf dash dl 1
+
+set pm3d map
+set contour surface
+set cntrparam levels discr 10 
+set samples 50
+set isosamples 50 
+
+set palette maxcolors 8
+set palette defined(\
+0 0.2314 0.2980 0.7529,\
+0.125000 0.384300 0.509800 0.917600,\
+0.250000 0.552900 0.690200 0.996100,\
+0.375000 0.721600 0.815700 0.976500,\
+0.500000 0.866700 0.866700 0.866700,\
+0.625000 0.960800 0.768600 0.678400,\
+0.750000 0.956900 0.603900 0.482400,\
+0.875000 0.870600 0.376500 0.302000,\
+1 0.7059 0.0157 0.1490\
+)
+
+set cbtics 0.6
+set mcbtics 8
+#set cblabel 'distance'
+#unset cbtics
+set cbrange [0:0.6]
+set format cb "%.1f"
+
+#set xrange[0:3]
+#set yrange[0:3]
+
+set xlabel 't [s]'
+set ylabel 't [s]'
+
+# evil manual meddling with axes tics
+set xtics ("0" 0, "500" 50, "1000" 100, "1500" 150, "2000" 199)
+set ytics ("0" 0, "500" 50, "1000" 100, "1500" 150, "2000" 199)
+#set mxtics 2
+#set mytics 2
+
+set size square
+
+
+set out 'myDistMatrix.tex'
+splot 'myMatrix.txt' matrix using ($1):($2):(1-$3) with image notitle

tutorials/rStatAnalysis/laminarFlowOvercylinder/run

@@ -0,0 +1,12 @@
+#!/bin/bash
+#
+# run the simulation
+
+rm -rf 0
+cp -r 0.org 0
+
+# initialise solution
+potentialFoam 2>&1 | tee log.potentialFoam 2>&1
+
+# run simulation
+pimpleFoam 2>&1 | tee log.pimpleFoam 2>&1

tutorials/rStatAnalysis/laminarFlowOvercylinder/system/blockMeshDict

@@ -0,0 +1,145 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \    /   O peration     | Version:  2.1.x                                 |
+|   \  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+	version		2.0;
+	format		ascii;
+	class		dictionary;
+	object		blockMeshDict;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+
+
+
+convertToMeters 1.0;
+
+
+vertices
+(
+	( 0.0 -0.25 0.0 )	 // x0y0z0 = 0
+	( 0.0 -0.25 0.01 )	 // x0y0z1 = 1
+	( 0.0 0.25 0.0 )	 // x0y1z0 = 2
+	( 0.0 0.25 0.01 )	 // x0y1z1 = 3
+	( 0.15000000000000002 -0.25 0.0 )	 // x1y0z0 = 4
+	( 0.15000000000000002 -0.25 0.01 )	 // x1y0z1 = 5
+	( 0.15000000000000002 0.25 0.0 )	 // x1y1z0 = 6
+	( 0.15000000000000002 0.25 0.01 )	 // x1y1z1 = 7
+	( 0.65 -0.25 0.0 )	 // x2y0z0 = 8
+	( 0.65 -0.25 0.01 )	 // x2y0z1 = 9
+	( 0.65 0.25 0.0 )	 // x2y1z0 = 10
+	( 0.65 0.25 0.01 )	 // x2y1z1 = 11
+	( 1.5 -0.25 0.0 )	 // x3y0z0 = 12
+	( 1.5 -0.25 0.01 )	 // x3y0z1 = 13
+	( 1.5 0.25 0.0 )	 // x3y1z0 = 14
+	( 1.5 0.25 0.01 )	 // x3y1z1 = 15
+
+	( 0.41767766952966373 0.017677669529663688 0.0 )	 // a0h0 = 16
+	( 0.3823223304703363 0.01767766952966369 0.0 )	 // a1h0 = 17
+	( 0.3823223304703363 -0.017677669529663688 0.0 )	 // a2h0 = 18
+	( 0.41767766952966373 -0.01767766952966369 0.0 )	 // a3h0 = 19
+	( 0.41767766952966373 0.017677669529663688 0.01 )	 // a0h1 = 20
+	( 0.3823223304703363 0.01767766952966369 0.01 )	 // a1h1 = 21
+	( 0.3823223304703363 -0.017677669529663688 0.01 )	 // a2h1 = 22
+	( 0.41767766952966373 -0.01767766952966369 0.01 )	 // a3h1 = 23
+);
+
+
+
+
+blocks
+(
+    hex ( 0 4 6 2 1 5 7 3 )   ( 6 20 1 ) simpleGrading ( 1 1 1 )  // 0   ['x0y0z0', 'x1y0z0', 'x1y1z0', 'x0y1z0']
+    hex ( 8 12 14 10 9 13 15 11 )   ( 34 20 1 ) simpleGrading ( 1 1 1 )  // 1   ['x2y0z0', 'x3y0z0', 'x3y1z0', 'x2y1z0']
+// cylinder block
+    hex ( 16 10 6 17 20 11 7 21 )   ( 18 20 1 ) simpleGrading ( 1 1 1 )  // 2   ['a0h0', 'b0h0', 'b1h0', 'a1h0']
+    hex ( 17 6 4 18 21 7 5 22 )   ( 18 20 1 ) simpleGrading ( 1 1 1 )  // 3   ['a1h0', 'b1h0', 'b2h0', 'a2h0']
+    hex ( 18 4 8 19 22 5 9 23 )   ( 18 20 1 ) simpleGrading ( 1 1 1 )  // 4   ['a2h0', 'b2h0', 'b3h0', 'a3h0']
+    hex ( 19 8 10 16 23 9 11 20 )   ( 18 20 1 ) simpleGrading ( 1 1 1 )  // 5   ['a3h0', 'b3h0', 'b0h0', 'a0h0']
+);
+
+
+
+
+edges
+(
+    arc 16 17 ( 0.4 0.025 0.0 )// cyl block
+    arc 17 18 ( 0.375 3.061616997868383e-18 0.0 )// cyl block
+    arc 18 19 ( 0.4 -0.025 0.0 )// cyl block
+    arc 19 16 ( 0.42500000000000004 -6.123233995736766e-18 0.0 )// cyl block
+    arc 20 21 ( 0.4 0.025 0.01 )// cyl block
+    arc 21 22 ( 0.375 3.061616997868383e-18 0.01 )// cyl block
+    arc 22 23 ( 0.4 -0.025 0.01 )// cyl block
+    arc 23 20 ( 0.42500000000000004 -6.123233995736766e-18 0.01 )// cyl block
+);
+
+
+
+
+boundary
+(
+
+inlet
+{
+  type patch;
+  faces
+  (
+    ( 0 2 3 1 )  // ['x0y0z0', 'x0y1z0', 'x0y1z1', 'x0y0z1']
+  );
+}
+
+
+outlet
+{
+  type patch;
+  faces
+  (
+    ( 12 14 15 13 )  // ['x3y0z0', 'x3y1z0', 'x3y1z1', 'x3y0z1']
+  );
+}
+
+
+cylinder
+{
+  type wall;
+  faces
+  (
+    ( 16 17 21 20 )  // ['a0h0', 'a1h0', 'a1h1', 'a0h1']
+    ( 17 18 22 21 )  // ['a1h0', 'a2h0', 'a2h1', 'a1h1']
+    ( 18 19 23 22 )  // ['a2h0', 'a3h0', 'a3h1', 'a2h1']
+    ( 19 16 20 23 )  // ['a3h0', 'a0h0', 'a0h1', 'a3h1']
+  );
+}
+
+
+sides
+{
+  type patch;
+  faces
+  (
+    ( 0 4 5 1 )  // ['x0y0z0', 'x1y0z0', 'x1y0z1', 'x0y0z1']
+    ( 2 6 7 3 )  // ['x0y1z0', 'x1y1z0', 'x1y1z1', 'x0y1z1']
+    ( 4 8 9 5 )  // ['x1y0z0', 'x2y0z0', 'x2y0z1', 'x1y0z1']
+    ( 6 10 11 7 )  // ['x1y1z0', 'x2y1z0', 'x2y1z1', 'x1y1z1']
+    ( 8 12 13 9 )  // ['x2y0z0', 'x3y0z0', 'x3y0z1', 'x2y0z1']
+    ( 10 14 15 11 )  // ['x2y1z0', 'x3y1z0', 'x3y1z1', 'x2y1z1']
+  );
+}
+
+);
+
+
+
+
+mergePatchPairs
+(
+);
+
+
+
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

tutorials/rStatAnalysis/laminarFlowOvercylinder/system/controlDict

@@ -0,0 +1,90 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  2.1.x                                 |
+|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+    version     2.0;
+    format      ascii;
+    class       dictionary;
+    location    "system";
+    object      controlDict;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+application     twoPhaseEulerFoam;
+
+startFrom       latestTime;
+
+startTime       0.1;
+
+stopAt          endTime;
+//stopAt          writeNow;
+
+endTime         2000;
+
+deltaT          1.0e-2;
+
+writeControl    adjustableRunTime;
+
+writeInterval   10;
+
+purgeWrite      0;
+
+writeFormat     binary;
+
+writePrecision  6;
+
+writeCompression uncompressed;
+
+timeFormat      general;
+
+timePrecision   6;
+
+runTimeModifiable yes;
+
+adjustTimeStep  yes;
+
+maxCo           0.25;
+
+maxDeltaT       1;
+
+
+functions
+{
+	forces
+    {
+        type        forceCoeffs;
+        functionObjectLibs ( "libforces.so" );
+        
+        writeControl    timeStep;
+        writeInterval   1;
+        
+        patches
+        (
+            cylinder
+        );
+		directForceDensity no;
+
+        pName       p;
+        UName       U;
+        rhoName     rhoInf;
+        //log         true;
+        rhoInf      994.5;
+        rho         rhoInf;
+        CofR        ( 0 0 0 );
+        liftDir     ( 0 1 0 );
+        dragDir     ( 1 0 0 );
+        pitchAxis   ( 0 0 1 );
+        magUInf     10.0;
+        lRef        0.04;
+        Aref        0.0157;
+        Aref1        0.004;
+        rhoRef      994.5;
+    }
+}
+
+// ************************************************************************* //

tutorials/rStatAnalysis/laminarFlowOvercylinder/system/fvSchemes

@@ -0,0 +1,62 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  2.3.0                                 |
+|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+    version     2.0;
+    format      ascii;
+    class       dictionary;
+    location    "system";
+    object      fvSchemes;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+ddtSchemes
+{
+    default         Euler;
+}
+
+gradSchemes
+{
+    default         Gauss linear;
+    grad(U)			cellLimited Gauss linear 1;
+}
+
+divSchemes
+{
+    default         none;
+    div(phi,U)      				bounded Gauss linearUpwindV grad(U);
+    
+    
+    div((nuEff*dev2(T(grad(U)))))           Gauss linear;
+    div((nuEff*dev(T(grad(U))))) 			Gauss linear;
+}
+
+laplacianSchemes
+{
+    default         Gauss linear limited corrected 0.33;
+}
+
+interpolationSchemes
+{
+    default         linear;
+}
+
+snGradSchemes
+{
+    default			corrected;
+}
+
+fluxRequired
+{
+    default         no;
+    p               ;
+    Phi             ;
+}
+
+
+// ************************************************************************* //

tutorials/rStatAnalysis/laminarFlowOvercylinder/system/fvSolution

@@ -0,0 +1,105 @@
+/*--------------------------------*- C++ -*----------------------------------*\
+| =========                 |                                                 |
+| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
+|  \\    /   O peration     | Version:  2.3.0                                 |
+|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
+|    \\/     M anipulation  |                                                 |
+\*---------------------------------------------------------------------------*/
+FoamFile
+{
+    version     2.0;
+    format      ascii;
+    class       dictionary;
+    location    "system";
+    object      fvSolution;
+}
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+solvers
+{
+    p
+    {
+        solver          GAMG;
+        tolerance       1e-08;
+        relTol          0.0001;
+        maxIter			100;
+        smoother        DIC;
+        nPreSweeps      1;
+        nPostSweeps     2;
+        nFinestSweeps   2;
+        scaleCorrection	true;
+        directSolveCoarsestLevel	false;
+        cacheAgglomeration true;
+        nCellsInCoarsestLevel 330;
+        agglomerator    faceAreaPair;
+        mergeLevels     1;
+    }
+
+    pFinal
+    {
+        $p;
+        relTol          0;
+    }
+
+    U
+    {
+        solver          PBiCG;
+        preconditioner  DILU;
+        tolerance       1e-08;
+        relTol          0.001;
+    }
+    
+    /*U
+    {
+        type            coupled;
+        solver          PBiCICG;
+        preconditioner  DILU;
+        tolerance       (1e-08 1e-08 1e-08);
+        relTol          (0 0 0);
+    }*/
+    
+    UFinal
+    {
+        $U;
+    }
+    
+    Phi
+    {
+        solver          PCG;
+        preconditioner  DIC;
+        tolerance       1e-08;
+        relTol          0.001;
+    }
+}
+
+PIMPLE
+{
+    nOuterCorrectors    10;
+    nCorrectors         1;
+    nNonOrthogonalCorrectors 1;
+    
+    residualControl
+    {
+        U
+        {
+            relTol          0;
+            tolerance       0.00001;
+        }
+    }
+}
+
+relaxationFactors
+{
+    /*equations
+    {
+        p						0.4;
+        U						0.6;		
+    }*/
+}
+
+potentialFlow
+{
+    nNonOrthogonalCorrectors 20;
+}
+
+// ************************************************************************* //